Definitions

Restricted 3-body problem

N-body problem

The n-body problem is the problem of finding, given the initial positions, masses, and velocities of n bodies, their subsequent motions as determined by classical mechanics, i.e., Newton's laws of motion and Newton's law of gravity.

Mathematical formulation of the n-body problem

The general n-body problem of celestial mechanics is an initial-value problem for ordinary differential equations. Given initial values for the positions mathbf{q}_j(0) and velocities dotmathbf{q}_j(0) of n particles (j = 1,...,n) with mathbf{q}_j(0) neq mathbf{q}_k(0) for all mutually distinct j and k , find the solution of the second order system

m_j ddot{mathbf{q}}_j = gamma sumlimits_{kneq j } frac{m_j m_k(mathbf{q}_k-mathbf{q}_j)}{|mathbf{q}_k-mathbf{q}_j|^3}, j=1,ldots,n qquad qquad qquad (1)

where m_1,m_2,ldots m_n are constants representing the masses of n point-masses and mathbf{q}_1,mathbf{q}_2,ldots,mathbf{q}_n are 3-dimensional vector functions of the time variable t, describing the positions of the point masses. This equation is simply _law_of_acceleration; the left-hand side is the mass times acceleration for the jth particle, whereas the right-hand side is the sum of the forces on that particle. The forces are assumed here to be gravitational and given by Newton's law of universal gravitation; thus, they are proportional to the masses involved, and vary as the inverse square of the distance between the masses. The power of three in the denominator is correct, since it balances the vector difference in the numerator, which is necessary to specify the direction of the force.

For n = 2, the problem was completely solved by Johann Bernoulli (see Two-body problem below).

General considerations: solving the n-body problem

In the physical literature about the n-body problem (n ≥ 3), sometimes reference is made to the impossibility of solving the n-body problem. However one has to be careful here, as this applies to the method of first integrals (compare the theorems by Abel and Galois about the impossibility of solving algebraic equations of degree five or higher by means of formulas only involving roots).

The n-body problem contains 6n variables, since each point particle is represented by three space (displacement) and three velocity components. First integrals (for ordinary differential equations) are functions that remain constant along any given solution of the system, the constant depending on the solution. In other words, integrals provide relations between the variables of the system, so each scalar integral would normally allow the reduction of the system's dimension by one unit. Of course, this reduction can take place only if the integral is an algebraic function not very complicated with respect to its variables. If the integral is transcendent the reduction cannot be performed.

The n-body problem has 10 independent algebraic integrals

  1. three for the center of mass
  2. three for the linear momentum
  3. three for the angular momentum
  4. one for the energy.

This allows the reduction of variables to 6n − 10. The question at that time was whether there exist other integrals besides these 10. The answer was given in 1887 by H. Bruns.

Theorem (First integrals of the n-body problem) The only linearly independent integrals of the n-body problem, which are algebraic with respect to q, p and t are the 10 described above.

(This theorem was later generalised by Poincaré). These results however do not imply that there does not exist a general solution of the n-body problem or that the perturbation series (Linstedt series) diverges. Indeed Sundman provided such a solution by means of convergent series. (See Sundman's theorem for the 3-body problem).

Two-body problem

If the common center of mass of the two bodies is considered to be at rest, each body travels along a conic section which has a focus at the center of mass of the system (in the case of a hyperbola: the branch at the side of that focus). The two conics will be in the same plane. The type of conic (circle, ellipse, parabola or hyperbola) is determined by finding the sum of the combined kinetic energy of two bodies and the potential energy when the bodies are far apart. (This potential energy is always a negative value; energy of rotation of the bodies about their axes is not counted here).

  • If the sum of the energies is negative, then they both trace out ellipses.
  • If the sum of both energies is zero, then they both trace out parabolas. As the distance between the bodies tends to infinity, their relative speed tends to zero.
  • If the sum of both energies is positive, then they both trace out hyperbolas. As the distance between the bodies tends to infinity, their relative speed tends to some positive number.

Note: The fact that a parabolic orbit has zero energy arises from the assumption that the gravitational potential energy goes to zero as the bodies get infinitely far apart. One could assign any value (e.g. 42 joules) to the potential energy in the state of infinite separation. That state is assumed to have zero potential energy (i.e. 0 joules) by convention.

See also Kepler's first law of planetary motion.

Three-body problem

For n ≥ 3 very little is known about the n-body problem. The case n = 3 was most studied, for many results can be generalised to larger n. The first attempts to understand the 3-body problem were quantitative, aiming at finding explicit solutions.

  • In 1767 Euler found the collinear periodic orbits, in which three bodies of any masses move such that they oscillate along a rotation line.
  • In 1772 Lagrange discovered some periodic solutions which lie at the vertices of a rotating equilateral triangle that shrinks and expands periodically. Those solutions led to the study of central configurations , for which ddot q=kq for some constant k>0 .

The three-body problem is much more complicated; its solution can be chaotic. A major study of the Earth-Moon-Sun system was undertaken by Charles-Eugène Delaunay, who published two volumes on the topic, each of 900 pages in length, in 1860 and 1867. Among many other accomplishments, the work already hints at chaos, and clearly demonstrates the problem of so-called "small denominators" in perturbation theory.

The restricted three-body problem assumes that the mass of one of the bodies is negligible; the circular restricted three-body problem is the special case in which two of the bodies are in circular orbits (approximated by the Sun-Earth-Moon system and many others). For a discussion of the case where the negligible body is a satellite of the body of lesser mass, see Hill sphere; for binary systems, see Roche lobe; for another stable system, see Lagrangian point.

The restricted problem (both circular and elliptical) was worked on extensively by many famous mathematicians and physicists, notably Lagrange in the 18th century and Poincaré at the end of the 19th century. Poincaré's work on the restricted three-body problem was the foundation of deterministic chaos theory. In the circular problem, there exist five equilibrium points. Three are collinear with the masses (in the rotating frame) and are unstable. The remaining two are located on the third vertex of both equilateral triangles of which the two bodies are the first and second vertices. This may be easier to visualize if one considers the more massive body (e.g., Sun) to be "stationary" in space, and the less massive body (e.g., Jupiter) to orbit around it, with the equilibrium points maintaining the 60 degree-spacing ahead of and behind the less massive body in its orbit (although in reality neither of the bodies is truly stationary; they both orbit the center of mass of the whole system). For sufficiently small mass ratio of the primaries, these triangular equilibrium points are stable, such that (nearly) massless particles will orbit about these points as they orbit around the larger primary (Sun). The five equilibrium points of the circular problem are known as the Lagrange points.

King Oscar II Prize about the solution for the n-body problem

The problem of finding the general solution of the n-body problem was considered very important and challenging. Indeed in the late 1800s King Oscar II of Sweden, advised by Gösta Mittag-Leffler, established a prize for anyone who could find the solution to the problem. The announcement was quite specific:

Given a system of arbitrarily many mass points that attract each according to Newton's law, under the assumption that no two points ever collide, try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the series converges uniformly.

In case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was finally awarded to Poincaré, even though he did not solve the original problem. (The first version of his contribution even contained a serious error; for details see the article by Diacu). The version finally printed contained many important ideas which led to the theory of chaos. The problem as stated originally was finally solved by Karl Fritiof Sundman for n=3.

Sundman's theorem for the 3-body problem

In 1912, the Finnish mathematician Karl Fritiof Sundman proved that there exists a series solution in powers of t^{1/3} for the 3-body problem. This series is convergent for all real t, except initial data which correspond to zero angular momentum. However these initial data are not generic since they have Lebesgue measure zero.

An important issue in proving this result is the fact that the radius of convergence for this series is determined by the distance to the nearest singularity. Therefore it is necessary to study the possible singularities of the 3-body problems. As it will be briefly discussed in the next section, the only singularities in the 3-body problem are

  1. binary collisions
  2. triple collisions.

Now collisions, whether binary or triple (in fact of arbitrary order), are somehow improbable since it has been shown that they correspond to a set of initial data of measure zero. However there is no criterion known to be put on the initial state in order to avoid collisions for the corresponding solution. So Sundman's strategy consisted of the following steps:

  1. He first was able, using an appropriate change of variables, to continue analytically the solution beyond the binary collision, a process known as regularization .
  2. He then proved that triple collisions only occur when the angular momentum c vanishes. By restricting the initial data to mathbf{c}neq 0 he removed all real singularities from the transformed equations for the 3-body problem.
  3. The next step consisted in showing that if mathbf{c}neq 0 then not only can there be no triple collision, but the system is strictly bounded away from a triple collision. This implies, by using the Cauchy existence theorem for differential equations, that there are no complex singularities in a strip (depending on the value of c) in the complex plane centered around the real axis.
  4. The last step is then to find a conformal transformation which maps this strip into the unit disc. For example if s=t^{1/3} (the new variable after the regularization) and if |mathop{text{Im}} , s| leq beta then this map is given by

sigma = frac{e^{pi s/(2beta)} - 1}{e^{pi s/(2beta) }+1}.

This finishes the proof of Sundman's theorem. Unfortunately the corresponding convergent series converges very slowly. That is, getting the value to any useful precision requires so many terms that his solution is of little practical use.

The global solution of the n-body problem

In order to generalise Sundman's result for the case n>3 (or n=3 and c=0) one has to face two obstacles:

  1. As it has been shown by Siegel, that collisions which involve more than 2 bodies cannot be regularised analytically, hence Sundman's regularization cannot be generalised.
  2. The structure of singularities is more complicated in this case, other types of singularities may occur.

Finally Sundman's result was generalised to the case of n>3 bodies by Q. Wang in the 1990s. Since the structure of singularities is more complicated, Wang had to leave out completely the questions of singularities. The central point of his approach is to transform, in an appropriate manner, the equations to a new system, such that the interval of existence for the solutions of this new system is [0,infty) .

Singularities of the n-body problem

There can be two types of singularities of the n-body problem:

  • collisions of one, two or n particles, but for which q(t) remains finite.
  • singularities in which a collapse does not occur, but q(t) does not remain finite. The latter one are called no-collisions singularities. Their existence has been conjectured for n > 3 by Painlevé (see Painlevé's conjecture). Examples of this behaviour have been constructed by Xia and Gerver.

See also

References

  • Diacu, F.: The solution of the n-body Problem, The Mathematical Intelligencer,1996,18,p.66–70
  • Mittag-Leffler, G.: The n-body problem (Price Announcement), Acta Matematica, 1885/1886,7
  • Saari, D.: A visit to the Newtonian n-body Problem via Elementary Complex Variables, American Mathematical Monthly, 1990, 89, 105–119
  • Wang, Qiudong: The global solution of the n-body problem (Celestial Mechanics and Dynamical Astronomy (ISSN 0923-2958), vol. 50, no. 1, 1991, p. 73–88., URI retrieved on 2007-05-05)
  • Sundman, K. E.: ''Memoire sur le probleme de trois corps, Acta Mathematica 36 (1912): 105–179.
  • Hagihara, Y: Celestial Mechanics. (Vol I and Vol II pt 1 and Vol II pt 2.) MIT Press, 1970.
  • Boccaletti, D. and Pucacco, G.: Theory of Orbits (two volumes). Springer-Verlag, 1998.

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